An algebraic approach to Coulomb scattering in <i>N</i> dimensions

Rasmussen, William O.; Salamó, S.

doi:10.1063/1.524198

Cited by 15 publications

(26 citation statements)

References 9 publications

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“…In fact, such a generalization should go back to the earlier works by Appel, Fock, Bargmann, Sommerfeld et al [16][17][18][19], the notes left by Bateman edited by Erdélyi in 1950s [20] and others [21]. Most of them paid more attention to the harmonic oscillator [13-15, 22, 23] than hydrogen atom [24][25][26][27][28][29]. Following Louck's work, de Broglie and his collaborators [30] proposed the generating bases as the hyperspherical harmonics to analyze the higher dimensional harmonic oscillator and molecular vibration.…”

Section: Introduction 1 Basic Reviewmentioning

confidence: 99%

Wave Equations in Higher Dimensions

Dong

2011

223

153

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No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. PrefaceThis work will introduce the wave equations in higher dimensions at an advanced level addressing students of physics, mathematics and chemistry. The aim is to put the mathematical and physical concepts and techniques like the wave equations, group theory, generalized hypervirial theorem, the Levinson theorem, exact and proper quantization rules related to the higher dimensions at the reader's disposal. For this purpose, we attempt to provide a comprehensive description of the wave equations including the non-relativistic Schrödinger equation, relativistic Dirac and Klein-Gordon equations in higher dimensions and their wide applications in quantum mechanics which complements the traditional coverage found in the existing quantum mechanics textbooks. Related to this field are the quantum mechanics and group theory. In fact, the author's driving force has been his desire to provide a comprehensive review volume that includes some new and significant results about the wave equations in higher dimensions drawn from the teaching and research experience of the author since the literature is inundated with scattered articles in this field and to pave the reader's way into this territory as rapidly as possible. We have made the effort to present the clear and understandable derivations and include the necessary mathematical steps so that the intelligent and diligent reader is able to follow the text with relative ease, in particular, when mathematically difficult material is presented. The author also embraces enthusiastically the potential of the LaTeX typesetting language to enrich the presentation of the formulas as to make the logical pattern behind the mathematics more transparent. In addition, any suggestions and criticism to improve the text are most welcome. It should be pointed out that the main effort to follow the text and master the material is left to the reader even though this book makes an effort to serve the reader as much as was possible for the author. This book starts out in Chap. 1 with a comprehensive review for the wave equations in higher dimensions and builds on this to introduce in Chap. 2 the fundamental theory about the SO(N ) group to be used in the successive Chaps. 3-5 including the non-relativistic Schrödinger equation, relativistic Dirac and Klein-Gordon equations. As important applications in non-relativistic quantum mechanics, from Chap. 6 to Chap. 12, we shall apply the theories proposed in Part II to study some vii viii Preface important quantum systems such as the harmonic oscillator, Coulomb potential, the Levinson theorem, generalized hypervirial theorem, exact and proper...

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Section: Introduction 1 Basic Reviewmentioning

confidence: 99%

Wave Equations in Higher Dimensions

Dong

2011

223

153

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“…The connection between the n-dimensional Kepler-Coulomb and harmonic oscillator and Lie algebras su(n) and so(n + 1) have been central to the development of algebraic techniques in physics and study of exactly solvable models [34,6,59,92,5,64,91,4,83,65,10,7,35,56]. They also motivated application of finite dimensional uunitary representations and Casimir operators and development of algebraic methods to study Hamiltonians.…”

Section: Introductionmentioning

confidence: 99%

Higher order superintegrability, Painlevé transcendents and representations of polynomial algebras

Marquette

2019

J. Phys.: Conf. Ser.

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In recent years, progress toward the classification of superintegrable systems with higher order integrals of motion has been made. In particular, a complete classification of all exotic potentials with a third or a fourth order integrals, and allowing separation of variables in Cartesian coordinates. All doubly exotic potentials with a fifth order integral have also been completely classified. It has been demonstrated how the Chazy class of third order differential equations plays an important role in solving determining equations. Moreover, taking advantage of various operator algebras defined as Abelian, Heisenberg, Conformal and Ladder case of operator algebras, we re-derived these models. These new techniques also provided further examples of superintegrable Hamiltonian with integrals of arbitrary order. It has been conjectured that all quantum superintegrable potentials that do not satisfy any linear equation satisfy nonlinear equations having the Painlevé property. In addition, it has been discovered that their integrals naturally generate finitely generated polynomial algebras and the representations can be exploited to calculate the energy spectrum. For certain very interesting cases associated with exceptional orthogonal polynomials, these algebraic structures do not allow to calculate the full spectrum and degeneracies. It has been demonstrated that alternative sets of integrals which can be build and used to provide a complete solution. This this allow to make another conjecture i.e. that higher order superintegrable systems can be solved algebraically, they require alternative set of integrals than the one provided by a direct approach.consisted in obtaining the determining equations (N=3) from a third order integral [48] and classifying all Hamiltonians allowing separation of variable in Cartesian coordinates with a third order integrals [49]. A connection with Painlevé transcendents was also established for the first time and demonstrated the significance of studying higher order superintegrability.Much recently, it has been recognized how the search can be narrowed to exotic potentials i.e. the potential do not satisfy any linear differential equation. All exotic potentials for integrals of fourth order (N=4) [67] and doubly exotic systems with integral of fifth order (N=5) [2] were obtained. These potentials can be rewritten in term of the first, second, third, fourth and fifth Painlevé transcendents. Superintegrable systems allowing separation of variables in parabolic coordinates for N=3 [88], as well as systems allowing separation of variables in polar coordinates for N=3 [94] and N=4 [29,30]. Models were obtained in terms of the sixth Painlevé transcendents. Some work have been done in regard of the case of Nth order integrals related to superintegrable systems with separation of variables in Cartesian coordinates [66] and [89]. In particular various alternative form for the integrals of motion have been obtained. Similar work for Nth order integrals have also been done in regard of polar coo...

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“…subalgebra chain so(N + 1) ⊃ so(N) ⊃ ... ⊃ so (2) ) to define appropriate quantum numbers [3,12,5,6,7,8]. Another derivation consist in using higher order Casimir operators.…”

Section: This System Has Integrals Of Motion Given By the Runge-lenz mentioning

confidence: 99%

“…The most known examples whose symmetry algebras are Lie algebras generated by integrals of motion are N-dimensional hydrogen atom and harmonic oscillator. See [3,12,5,6,7,8] for systems with so(N + 1) symmetry and [9,10,11,12,13] for those with su(N) symmetry.…”

Section: Introductionmentioning

confidence: 99%

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Quadratic algebra structure and spectrum of a new superintegrable system inN-dimension

Hoque¹,

Marquette²,

Zhang³

2015

J. Phys. A: Math. Theor.

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We introduce a new superintegrable Kepler-Coulomb system with non-central terms in N -dimensional Euclidean space. We show this system is multiseparable and allows separation of variables in hyperspherical and hyperparabolic coordinates. We present the wave function in terms of special functions. We give a algebraic derivation of spectrum of the superintegrable system. We show how the so(N + 1) symmetry algebra of the N -dimensional Kepler-Coulomb system is deformed to a quadratic algebra with only 3 generators and structure constants involving a Casimir operator of so(N − 1) Lie algebra. We construct the quadratic algebra and the Casimir operator. We show this algebra can be realized in terms of deformed oscillator and obtain the structure function which yields the energy spectrum.

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An algebraic approach to Coulomb scattering in N dimensions

Abstract: Using purely algebraic techniques, based on the larger symmetry group of the Kepler problem, the phase shifts and the scattering amplitude for Coulomb scattering in N dimensions are derived.

Cited by 15 publications

References 9 publications

Wave Equations in Higher Dimensions

Wave Equations in Higher Dimensions

Higher order superintegrability, Painlevé transcendents and representations of polynomial algebras

Quadratic algebra structure and spectrum of a new superintegrable system inN-dimension

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